Proof that the shape operator is a self adjoint application

Question

I am trying to prove that the shape operator in the first fundamental form is self adjoint. I have an application $\mathbb{X}: (u,v) \subset \mathbb{R}^2 \longrightarrow \mathbb{R}^3$ which determines a base of the tangent plane in a point $P$ of my surface, given by $\{ \mathbb{X}_u \mathbb{X}_v \}$, so I just have to proof that $\mathcal{F}_p(w_1) \cdot w_2 = w_1 \cdot \mathcal{F}_p(w_2)$ for $w_1, w_2$ being all the possible combinations between $\mathbb{X}_u, \mathbb{X}_v$. I am having trouble when $w_1 = \mathbb{X}_u$ and $w_2 = \mathbb{X}_v$, where: $$\mathcal{F}_p(\mathbb{X}_u)\cdot \mathbb{X}_v=-N_u\cdot \mathbb{X}_v=-\mathbb{X}_u \cdot N_v = \mathbb{X}_u \cdot \mathcal{F}_p(\mathbb{X}_v)$$ I don't understand why the second equality is true. Could someone please help me?

Answer 1

Following @Ben123's comments, I realized that my answer needs to be edited. I will explain things better this time.

Let $\overline{M}$ be a (semi-)Riemannian manifold, with metric denoted by $\bar{g}$ and let $M$ be a submanifold of $\overline{M}$, with induced metric denoted by $g$. We denote by $\overline{\nabla}$ and $\nabla$ the Levi-Civita connections on $\overline{M}$ and $M$ respectively. By definition, a Levi-Civita connection is torsion-free and compatible with the metric. Its existence and uniqueness is a theorem (which is sometimes known as the fundamental theorem of differential geometry).

Given a point $p \in M$, we note that the tangent space $T_p(\overline{M})$ of $\overline{M}$ at $p$ decomposes as: $$ T_p(\overline{M}) \simeq T_p(M) \oplus N_p, $$ (an orthogonal decomposition with respect to $\bar{g}_p$) using the inner product $\bar{g}_p$, where $T_p(M)$ denotes the tangent space of $M$ at $p$ and $N_p$ denotes the fiber of the normal bundle $N$ of $M$ as a submanifold of $\overline{M}$.

Given $X$, $Y \in T_p(M)$, we extend them to some local vector fields in some neighborhood $U$ of $p$ in $\overline{M}$, which are tangent to $M$ at $U \cap M$, we define $$ \Pi_p(X, Y) = \overline{\nabla}_X(Y)^\perp $$ to be the orthogonal (with respect to $\bar{g}$) projection of $\overline{\nabla}_X(Y)$ onto the fiber $N_p$ of the normal bundle. $\Pi(-,-)$ is called the second fundamental form. It is well defined as a local smooth section of $T^*(M) \otimes T^*(M) \otimes N$. In other words, the second fundamental form does not depend on the choices of extensions of $X$ and $Y$.

Also, given $X$ and $Y \in T_p(M)$ and $\xi \in N_p$, we define the shape operator $S_p^\xi \in End(T_p(M))$ by $$ g_p(S_p^\xi(X), Y) = \bar{g}_p(\Pi_p(X, Y), \xi). $$

Note that we have $$ g_p(S_p^\xi(X), Y) = \bar{g}_p(\overline{\nabla}_X(Y), \xi) = \bar{g}_p(\overline{\nabla}_Y(X) + [X, Y], \xi),$$ using the fact that $\overline{\nabla}$ is torsion-free. But $[X, Y] \in T_p(M)$, so it is orthogonal to $\xi$ (with respect to $\bar{g}_p$). We thus get $$ g_p(S_p^\xi(X), Y) = \bar{g}_p(\overline{\nabla}_Y(X), \xi) = g_p(X, S^\xi_p(Y)),$$ so that $S^\xi_p$ is indeed self-adjoint with respect to $g_p$.